The authors study the full system of Braginskii one-fluid transport equations with a self-consistent account of transport and dissipative phenomena in the screw-pinch geometry and derive exact self-similar solutions. The local and global plasma parameters-magnetic field, temperature and density profiles, magnetic Reynolds number, beta and pinch radius-are determined by a minimal set of external parameters, namely properly scaled magnitude of the applied axial field and an index related to the current time-evolution, and are computed as eigenfunctions and eigenvalues of the corresponding boundary value problem. The paramagnetic and diamagnetic properties of the axial field profiles are discussed in detail. Calculation of the m=1 growth rates demonstrates a relatively weak decrease of the maximum growth rate as the axial field amplitude increases although the range of unstable wavenumbers becomes substantially narrower. The solutions obtained describe plasma dynamics and profile structure in 'stabilized' pinch systems: Z-pinches with externally applied axial field and in ULQ pinches provided self-relaxation processes are suppressed.